Topic 7.10 Notes – Skills Focus: Selecting, Implementing, and Communicating Inference Procedures

Choosing the correct inference procedure

Every inference question boils down to three decisions:

1. What type of parameter?

   • Mean (μ) → quantitative data → use t procedures
   • Proportion (p) → categorical success/failure → use z procedures

   On the AP exam, you almost never know the population standard deviation, so means use t, not z.

2. How many samples or groups?

   • One sample
   • Two independent samples
   • Matched pairs (same subjects twice or naturally paired data)

3. Interval or test?

   • Estimate → confidence interval
   • Test a claim → significance test

If you can answer those three, the procedure picks itself.

The full set you’re responsible for

One Sample

• One-proportion z-interval / z-test
• One-sample t-interval / t-test

Two Independent Samples

• Two-proportion z-interval / z-test
• Two-sample t-interval / t-test

Matched Pairs

• Matched pairs t-interval / t-test
  • Analyze the differences and run a one-sample t on those differences.

More than two proportions

• Chi-square test (association or homogeneity)

Students lose points every year by confusing:

• Two-sample t vs. matched pairs t
• Two-proportion z vs. chi-square

If it’s exactly two proportions, use a two-proportion z. Save chi-square for 2+ categories.

Setting up inference with SPDC

Whether it’s a test or interval, your structure should always follow State, Plan, Do, Conclude.

1. State

Confidence interval

• Define the parameter in context. Example: $\mu$ = the true mean number of hours students at Central High sleep per night.

Significance test

• Write hypotheses in symbols:

$$H_0: \mu = 7$$

$$H_a: \mu < 7$$

• Null has equality.
• Hypotheses are about population parameters, never sample statistics.

2. Plan

You must check three conditions:

• Random
  Random sample or random assignment stated.

• Independent
  If sampling without replacement, verify the 10% condition.

• Normal / Large Counts

  • For means → population normal OR $n \ge 30$
  • For proportions:
    • Interval → $n\hat{p} \ge 10$ and $n(1-\hat{p}) \ge 10$
    • Test → $np_0 \ge 10$ and $n(1-p_0) \ge 10$

Notice the difference. Tests use $p_0$ in the condition. Intervals use $\hat{p}$.

3. Do

• Name the procedure (“two-sample t-test”).
• Use your calculator.
• Report:
  • Interval → the full interval.
  • Test → test statistic, df (for t), and p-value.

For example, if a one-sample t-test gives:

• $t = -2.14$
• $\text{df} = 24$
• $p = 0.021$

Write all three.

4. Conclude in context

For a test:

• If $p \le \alpha$ → reject $H_0$
• If $p > \alpha$ → fail to reject $H_0$

Example conclusion:

Since $p = 0.021 < 0.05$, we reject $H_0$. There is convincing evidence that the true mean hours of sleep for students at Central High is less than 7 hours per night.

Never say:

• “Accept $H_0$”
• “The null is false”
• A conclusion with no context

Interpreting a p-value correctly

A p-value is:

The probability of getting a sample statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true.

A strong AP-style interpretation includes:

• Probability language
• Refers to the sample statistic
• Mentions “as extreme or more extreme”
• States “assuming $H_0$ is true”

Example:

If the true mean sleep time is 7 hours, the probability of obtaining a sample mean this low or lower from a sample of 25 students is 0.021.

Common mistake: saying it’s the probability that $H_0$ is true. It is not.

Confidence intervals and tests are connected

Here’s how they relate:

Big connection:

If a hypothesized value is not inside a C% confidence interval, you would reject $H_0$ at $\alpha = 1 - C$.

Example: If 0 is not in a 95% CI for $\mu_1 - \mu_2$, that corresponds to rejecting at $\alpha = 0.05$.

Common traps

• Using z for means.
• Forgetting to check conditions before calculating.
• Mixing up independent vs. paired data.
• Comparing p-value to 0 instead of α.
• Writing a conclusion that ignores the alternative hypothesis.

When you feel stuck, quietly ask yourself:
Mean or proportion?
One, two, or paired?
Interval or test?

That decision tree solves most multiple-choice questions before you ever touch your calculator.